in inhomogeneous participating media author's personal copy

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Spectral Monte Carlo models for nongray radiation analysesin inhomogeneous participating media

Anquan Wang, Michael F. Modest *

Department of Mechanical and Nuclear Engineering, Pennsylvania State University, University Park, PA 16802, USA

Received 22 October 2006; received in revised form 3 February 2007Available online 16 April 2007

Abstract

A spectral line-by-line (LBL) method is developed for photon Monte Carlo simulations of radiation in participating media. The per-formance of the proposed method is compared with that of the stochastic full-spectrum k-distribution (FSK) method in both homoge-neous and inhomogeneous media, and in both traditional continuum media and media represented by stochastic particle fields, which arefrequently encountered in combustion simulations. By using random-number relations, the LBL method does its own spectral reorder-ing, in effect similar to the FSK reordering. Both LBL and FSK methods result in the same level of statistical errors for similar numbersof photon bundles. It is shown that the FSK approach is superior to the LBL approach in both memory and CPU efficiency in all testcases. However, the CPU efficiency is not prominent when spectral calculations are not the dominant source of overall CPU-time cost.Due to a lack of correlation of absorption coefficients in inhomogeneous media, the FSK approach may produce substantial errors,which are avoided by the LBL method. The LBL database can be constructed once and for all, while the FSK database has to be recon-structed every time the reference state changes. Thus, the LBL method is more suitable for quickly-evolving media.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Monte Carlo; Nongray; Participating media; Thermal radiation

1. Introduction

Radiation of participating media has great influence onheat transfer in many high-temperature applications, suchas combustion problems. A variety of radiative transfermodels has been developed to deal with radiation in partic-ipating media. Among them the photon Monte Carlo(PMC) method can handle problems of high complexitywith relative ease. Based on random-number relations, his-tories of energy bundles are traced to simulate the physicalprocesses, such as emission, scattering, absorption, etc.,and the result approaches the exact solution with diminish-ing statistical error with increasing numbers of energy bun-dles. Well-known disadvantages of this method are its highcomputational costs and statistical scatter. However, these

become less important with the rapid increase of computa-tional capacity of computers.

Early PMC studies often assumed gases to be gray, suchas Howell and Perlmutter’s work [1], which appears to bethe first application of PMC methods in radiative heattransfer. However, the gray-gas assumption may yieldunacceptable errors in many engineering applications, sincefor most gases the spectral variation of radiative propertiesis very strong. The first modern study on nongray gasesusing the PMC method was made by Modest [2] in 1992,who developed random-number relations based on a statis-tical narrow-band model to take spectral line structure intoaccount. Similar methods were employed in several otherPMC studies since then. Liu and Tiwari [3] used a statisti-cal narrow-band model with an exponential-tailed-inverseintensity distribution in the investigation of radiative dissi-pation inside the medium and net radiative wall heat fluxfor different temperature and concentration profiles in a1D problem. Cherkaoui et al. [4] used the statistical

0017-9310/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijheatmasstransfer.2007.02.018

* Corresponding author. Fax: +1 814 863 4848.E-mail address: [email protected] (M.F. Modest).

www.elsevier.com/locate/ijhmt

International Journal of Heat and Mass Transfer 50 (2007) 3877–3889

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narrow-band model in a 1D slab problem with prescribedtemperature fields and the reciprocity principle wasemployed to reduce the statistical error. Farmer andHowell [5,6] developed random-number relations basedon an exponential wide band absorption profile to treatnongray emission and absorption of carbon dioxide. Mostrecently, Wang et al. [7] incorporated the full-spectrum k-distribution (FSK) method, the most promising nongraymodel for deterministic solvers, into the PMC method inan artificial 1D problem. Tang and Brewster [8] used theline-by-line (LBL) method, the most accurate spectralmodel, combined with PMC to serve as benchmark forother wideband and narrow-band models in a 1D homoge-neous medium. In their work, rather than developing ran-dom-number relations to determine wavenumbers, spectralintegrations are used to give total radiative quantities. Nev-ertheless, to our knowledge, it is the only work combiningthe LBL and PMC methods.

Recently, the PMC method has been implemented inseveral CFD simulations of combustion, due to the contin-uous improvements in numerical methods and computercapabilities. Snegirev [9] incorporated a PMC method intoa CFD code with the weighted-sum-of-gray-gases (WSGG)model in his study of buoyant turbulent nonpremixedflames. Tesse et al. [10,11] developed their PMC methodsto model the radiative transfer in a turbulent sooty flame,in which a narrow-band correlated-k model was used forspectral integration. For a variety of combustion applica-tions, in which joint probability-density-function (PDF)methods [12,13] are used, the medium is represented by sta-tistical particles evolving with time, and traditional MonteCarlo ray-tracing methods developed for continuous mediaare no longer useful. Most recently, Wang and Modest

[14,15] developed several emission and absorption schemesfor discrete particle fields, which can be utilized to performPMC simulations in such media. The PMC method hasalso been applied in a variety of other participating media.For example, Xia et al. [16] developed a curved MonteCarlo method for radiative heat transfer in media with var-iable index of refraction. Ruan et al. [17] applied the MonteCarlo method in media with nongray absorbing–emitting-anisotropic scattering particles, and Mazumder and Kersch[18] employed a Monte Carlo scheme for thermal radiationin semiconductor processing applications.

The present paper proposes a spectral LBL approach,based on random-number relations, to be incorporatedinto PMC models for nongray radiation analyses in com-bustion gases. By using random-number relations, thewavenumbers with substantial emission are more likely tobe chosen. Therefore, it is expected that fewer wavenum-bers are required to yield accuracy comparable to the oneobtained from standard spectral integration. The LBLmethod is compared with the PMC/FSK approach in termsof CPU efficiency, memory requirements, accuracy, statis-tical errors, etc., using several sample calculations.

2. Theoretical development

2.1. Photon Monte Carlo method

Gas molecules in a combustion medium continuouslyemit energy in the form of photons into random directionsat distinct wavenumbers, which are determined by theenergy difference between the two quantized energy levelsof an emitting molecule before and after emission. Photonsmay be absorbed or scattered by other molecules during

Nomenclature

A boundary (wall) areaa nongray stretching factor in FSK methodg cumulative k-distributionI number of species; radiative intensityIb blackbody radiative intensityk absorption coefficient variableL cylinder lengthNray number of rays per runNwv number of wavenumbers (or g’s) per rayn local boundary surface normalp pressureq local radiative heat flux vector�q averaged boundary fluxR random number; cylinder radiuss position variable along ray pathT temperaturetcpu average CPU time per runx mole fractionx, y, z, r coordinates

Greek symbols

g wavenumberU scattering phase function/ gas state variablej absorption coefficientjp pressure-based Planck-mean absorption coeffi-

cientX0 solid angler Stefan–Boltzmann constant; standard deviationrs scattering coefficients optical thickness

Subscripts

0 reference stateg spectrali species index

3878 A. Wang, M.F. Modest / International Journal of Heat and Mass Transfer 50 (2007) 3877–3889

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transmission. Photon Monte Carlo (PMC) methodsdirectly simulate such processes by releasing representativephotons bundles (rays) into random directions, which aretraced until they are absorbed at certain points in the med-ium or escape from the domain. In traditional continuousmedia photon rays carry a fixed amount of energy andthe optical thickness they can travel through is determinedrandomly as

so ¼ lnð1=RsÞ; so ¼Z s�

0

jds; ð1Þ

where so is the predetermined optical thickness for a spe-cific ray, Rs is the random number uniformly distributedin [0,1), s is the position variable along the path and s* isthe location at which the ray is absorbed. Starting fromthe emission location, the optical thickness passed throughby a ray is accumulated during tracing until its correspond-ing so value is reached, then all the energy carried by theray is absorbed by the subvolume at the absorbing loca-tion. This method is called the standard Monte Carlo meth-od. If the medium is optically thin, most of the rays mayexit open boundaries without contributing to the statisticsin the medium. Similarly, if the medium is optically verythick, rays travel only very short distances without contrib-uting to statistics of elements far away. In both cases thestandard method becomes very inefficient. An alternative,the energy partitioning method, was proposed to alleviatesuch problem [19,20], in which the ray energy is absorbedgradually and partitioned to all the subvolumes it traversesuntil depletion or exiting. Both standard and energy parti-tioning methods can also be implemented in media repre-sented by statistical particles, in which the ray energy isabsorbed by a single particle when the optical thicknessreaches its predetermined value in the standard method,or the energy is distributed over all particles that the rayinteracts with in the energy partitioning method [14].

In continuous media emission points of rays are deter-mined randomly according to the emissive energy distribu-tion in the domain. In hot zones of combustion gases theconcentrations of carbon dioxide and water vapor tendto be high and strong emission is observed. Thus, hot zonesmust be represented by more rays than cold zones withweak emission, and the ray number density reflects theemissive energy distribution, which is achieved by con-structing random-number relations for emission locations[21]. By inverting such random-number relations, the emis-sion location can be determined as

x ¼ xðRxÞ; y ¼ yðRy ; xÞ; z ¼ zðRz; x; yÞ; ð2Þ

where Rx, Ry and Rz are independent random numbers. Inmedia represented by particle fields rays are emitted fromthe inside of particles and particle centers can be chosenas emission points if particles are small [14]. Since particlesmay emit different amounts of energy, the particle energymay be split into several rays in hot zones and several par-ticle energies may be combined together to emit a single ray

in cold zones, to constrain ray energy to a small range andminimize statistical error [15].

The determination of isotropic emission directions isidentical for both continuous media and media representedby particle fields, using random-number relations that emitrays into all directions with equal probability. For grayanalyses the Planck-mean absorption coefficient is used incalculations and no spectral modeling is required. For non-gray analyses wavenumbers carried by photon bundlesmust be randomly determined according to a certain ran-dom-number relation, which is the focus of the presentpaper. In standard simulations each photon bundle isassigned a single but different wavenumber. If the numberof rays required to resolve spectral variations is much lar-ger than the number of rays required to resolve spacialvariations in the problem, it may be advantageous to assignmultiple wavenumbers to each ray and thus trace fewerrays to reduce computational time. While equal energiesare initially given to each wavenumber, the requiredpath-length in ray-tracing is different for different wave-numbers due to the spectral variation of absorption coeffi-cients and/or the different predetermined optical thicknessfor each wavenumber. Therefore, the tracing of a photonbundle carrying multiple wavenumbers is not completeduntil its energies associated with all wavenumbers areabsorbed or it exits the domain.

For either continuous media or media represented byparticle fields, random-number relations can also be con-structed for scattering in conventional ways, if required[21].

2.2. Spectral random-number relations

The probability of the number of photons emitted in adifferential wavenumber interval dg is proportional to thePlanck function weighted by the spectral absorption coeffi-cient, i.e.,

Probabilityfg in dgg / jgIbg dg; ð3Þ

where g denotes the wavenumber, jg is the correspondinglocal spectral absorption coefficient and Ibg is the spectralblackbody intensity. Therefore, to simulate this emissionprocess of photons statistically for a given gas species i,the random-number relation for the emission wavenumberis derived as

Rg;i ¼R g

0 jg;iIbg dgR10 jg;iIbg dg

¼R g

0 jpg;iIbg dgR10 jpg;iIbg dg

¼ p

jp;irT 4

Z g

0

jpg;iIbg dg; ð4Þ

where Rg,i is a random number uniformly distributed in[0,1), jpg,i = jg,i/pi is the pressure-based spectral absorp-tion coefficient and pi is the partial pressure of species i;jp,i is the pressure-based Planck-mean absorption coeffi-cient, r is the Stefan–Boltzmann constant and T is thegas temperature. Since Eq. (4) is an implicit relation

A. Wang, M.F. Modest / International Journal of Heat and Mass Transfer 50 (2007) 3877–3889 3879

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between the random number and the wavenumber, it ismore convenient to tabulate this relation in a database toinvert random numbers into emission wavenumbers.

In practice one deals with gas mixtures. Consideringthat the absorption coefficient is additive,

jg ¼X

i

jg;i ¼X

i

jpg;ipi and

jpg ¼ jg=p ¼X

i

xijpg;i; ð5Þ

where xi = pi/p is the mole fraction of species i and p is thetotal pressure of the mixture, one can obtain the random-number relation for the gas mixture from Eq. (4) as

Rg ¼p

jprT 4

Z g

0

jpgIbg dg

¼ p

rT 4P

ixijp;i

Z g

0

Xi

xijpg;iIbg dg

¼ p

rT 4P

ixijp;i

Xi

xi

Z g

0

jpg;iIbg dg

¼X

i

xijp;iRg;i

,Xi

xijp;i: ð6Þ

Eq. (6) establishes a direct relation between the mixturerandom number Rg and species random numbers {Rg,i}.Since the relation between the mixture random numberRg and the corresponding wavenumber g is implicit, directinversion from Rg to g through Eq. (6) is impossible andmust be done by trial-and-error. For a given wavenumberg, the species random numbers {Rg,i} are first determined,followed by the calculation of Rg through Eq. (6). Forabsorption calculations, the desired mixture absorptioncoefficient jg at a given wavenumber can be directly calcu-lated from species pressure-based absorption coefficients{jpg,i} through Eq. (5). Therefore, a database tabulatingboth Rg,i–g and jpg,i–g relations of each species can be uti-lized to determine emission wavenumbers and absorptioncoefficients for the mixture, and such a database can beconstructed once and for all. If the total pressure is fixed,both the species random number, as in Eq. (4), and thepressure-based absorption coefficient are functions ofwavenumber, temperature and species concentration only,i.e.,

Rg;i ¼ fR;iðg; T ; xiÞ; jpg;i ¼ fj;iðg; T ; xiÞ;i ¼ 1; 2; . . . ; I ; ð7Þ

where I is the number of species. If self-broadening effectsof absorption coefficients are negligible, Rg,i and jpg,i arefunctions of temperature and wavenumber only. Thus, a3D interpolation scheme (2D if self-broadening is ne-glected) is sufficient for the database and the computationaleffort increases only linearly with increasing number of spe-cies. Fig. 1 shows the random number and correspondingabsorption coefficient distributions of a gas mixture in asmall spectral interval. Although the random number is a

monotonically increasing function, it has strongly varyinggradients even in such a small interval. A small error inrandom number may result in a significant deviation inabsorption coefficient. Therefore, common root-findingtechniques relying on smooth gradients, such as the New-ton–Raphson method, cannot be used here to invert ran-dom numbers. Instead, a bisectional search algorithmmay be employed in such a process since the Rg–g relationis monotonic, if the resolution of the random number data-base is sufficient to resolve those steps. Spectral scatteringcoefficients may also be added into the database if scatter-ing is considered.

In the FSK approach the oscillating absorption coeffi-cient spectrum, jg–g, is reordered into a well-behavedmonotonic k–g distribution, in which k is the absorptioncoefficient variable and g is the Planck-function-weightedcumulative density function of k. Correspondingly, byemploying the correlated-k assumption, the original radia-tive transfer equation (RTE) [21]

dIg

ds¼ jgðIbg � IgÞ � rsg Ig �

1

4p

Z4p

Igðs0ÞUðs; s0ÞdX0� �

ð8Þ

is mapped into g-space as [22]

dIg

ds¼ kðT 0;/; gÞ½aðT ; T 0; gÞIb � Ig�

� rs Ig �1

4p

Z4p

Igðs0ÞUðs; s0ÞdX0� �

; ð9Þ

where Ig is the radiative intensity to be solved in g-space,k(T0,/,g) is the full-spectrum k-distribution evaluated byreordering absorption coefficients at the local state /(including temperature, pressure and concentrations)weighted by the Planck function calculated at the reference

Wavenumber, cm-1A

bsor

ptio

n co

effic

ient

, cm

-12300 2301 2302 2303 2304 2305

0.01

0.02

0.03

Ran

dom

num

ber

0.47

0.48

0.49p= 1 bar, T= 1200 KxCO2=0.1,xH2O=0.2

Fig. 1. Random number and absorption coefficient distributions in asmall spectral interval (absorption coefficient data obtained from Wangand Modest [27]).


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temperature T0, and a(T,T0,g) is a nongray stretchingfunction reflecting the ratio of two FSK’s evaluated atthe same state /0 but using two different Planck-functiontemperatures, viz., local temperature T and reference tem-perature T0,

aðT ; T 0; gÞ ¼dgðT ;/0; k

�Þ=dk�

dgðT 0;/0; k�Þ=dk�

; k� ¼ kðT 0;/0; gÞ; ð10Þ

where k* corresponds to the given g in the k–g distributionevaluated at the reference state /0 and Planck-functiontemperature T0.

The first term in brackets on the right-hand side of Eq.(9) represents augmentation of Ig due to local emission andthe second term in brackets is the attenuation due to localabsorption. For the simulation of emission using MonteCarlo methods, the random-number relation for the deter-mination of a cumulative k-distribution value g, acting likean artificial wavenumber, is then derived as [7]

Rg ¼R g

0 kðT 0;/; gÞaðT ; T 0; gÞIb dgR 1

0kðT 0;/; gÞaðT ; T 0; gÞIb dg

¼R g

0kðT 0;/; gÞaðT ; T 0; gÞdgR 1

0 kðT 0;/; gÞaðT ; T 0; gÞdg; ð11Þ

where Rg is the random number and the upper limit g in thenumerator is the desired artificial wavenumber of emission.The denominator is the local Planck-mean absorption coef-ficient evaluated using the correlated-k assumption.

As shown in Eq. (10), the stretching function a(T,T0,g)does not depend on the k–g distribution at the local gasstate /. For truly correlated absorption coefficients, thelocal emission at state / = (p,T,x) can be evaluated exactlyusing the FSK method, i.e.,Z 1

0

jgð/ÞIbgðT Þdg ¼Z 1

0

kðT ;/; g�ÞIbðT Þdg�

¼Z 1

0

kðT 0;/; gÞaðT ; T 0; gÞIbðT Þdg;

ð12Þ

where g*(k) is the cumulative k-distribution for the localstate. However, in reality, absorption coefficients at differ-ent gas states are never truly correlated, i.e., the last part ofEq. (12) becomes an approximation. Modest and Zhang[23] observed a maximum error of 4% when they evaluatedradiative sources in combustion products. When includingradiation from CH4, the error was less than 5% everywhereexcept for the inlet region where large amounts of fuel ex-ist. In this region the error was below 10% except for fewsmall spots with large errors [23]. They also observed thatthe error of the FSK method can be as large as 26% forwall fluxes in a two-layer CO2–H2O–N2 mixture slabbounded by cold black walls, in which both temperatureand mole fractions have step changes in the slab [24].

Eq. (11) is an implicit relation and the random-numberinversion is conducted by a table-search method as well.

Since the process of assembling of FSK’s from absorptioncoefficients is fairly tedious, it is impractical to assembleFSK’s during ray-tracing. Instead, Rg, k and a should betabulated in a database at a given reference state for a setof temperatures and mole fractions before ray-tracing com-mences. For a given reference state /0, while the a-functionis a function of temperature and g only, both the randomnumber Rg and k are functions of temperature, g and spe-cies concentrations,

Rg ¼ fRgðg; T ; x1; x2; . . . ; xIÞ;k ¼ fkðg; T ; x1; x2; . . . ; xIÞ: ð13Þ

As a result, the interpolation scheme required by the FSKdatabase is (I + 2)-dimensional. As the number of species,I, increases, the interpolation algorithm becomes more andmore complicated and CPU cost increases exponentially.In addition, as pointed out by Modest and Zhang [23],for poorly correlated gases the reference state can have astrong effect on the accuracy of the FSK approach. Thus,the FSK database must be reconstructed from time to time,if the overall temperature and concentration fields undergosubstantial changes with time, which greatly affect the ref-erence state.

3. Construction of random-number databases

A line-by-line random-number database has been con-structed for water vapor and carbon dioxide, the two mostimportant combustion gases. The spectroscopic databaseHITEMP-2000 was used to calculate absorption coeffi-cients of water vapor, while CDSD-1000 was employedfor carbon dioxide, since it is considered to be more accu-rate than HITEMP [25,26]. To resolve the irregularity ofthe absorption coefficient across the spectrum, the absorp-tion coefficient must be evaluated at close enough spectrallocations. In the present work constant spectral spacing isused across the entire spectrum; the resolution is consid-ered fine enough if, when doubling the resolution, the errorof the narrow-band-mean absorption coefficient staysbelow 0.5% in major absorption bands across the entirespectrum [27]. For a given total pressure, the resolutionrequired depends on temperature and mole fraction. At atotal pressure of 1 bar, the finest resolution in the presentwork, 0.004 cm�1, is used for temperatures over 2000 Kat mole fractions less than 0.25, while the crudest,0.01 cm�1, is used for many other temperatures and molefractions. The absorption coefficient at a specific spectrallocation is the result of the overlap of nearby broadenedspectral lines. At the most frequently encountered pressurein practice (1 bar) Lorentz broadening is the dominant line-broadening mechanism. Since the extent of Lorentz broad-ening is inversely proportional to the square-root of tem-perature and additional spectral lines appear at highertemperature levels (‘‘hot lines”), temperature has greatinfluence on the spectral shapes of the absorption coeffi-cient. Thus, 23 temperatures equally-spaced between


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300 K and 2500 K are included in the database. In con-trast, mole fractions affect line shapes of pressure-basedabsorption coefficient only through self-broadening, andthis effect is fairly small for CO2, as shown in Fig. 2a, whichshows pressure-based absorption coefficients across a smallspectral interval at different mole fractions and a tempera-ture of 1200 K for CO2. However, the self-broadeningeffect is considerably larger for H2O, as shown in Fig. 2b,which implies that more mole fractions need to be includedin the database for H2O.

A homogeneous 1D gas slab problem, in which a mix-ture of nitrogen and CO2 or H2O is contained betweentwo parallel cold black walls, was investigated to evaluateself-broadening effects on radiation calculations. It wasseen that, for CO2, neglecting self-broadening resulted inan error of around 0.2% in terms of radiative fluxes for var-ious slab thicknesses. Therefore, a single mole fraction, say0.0, is sufficient for the database of CO2. However, the self-broadening effect is much stronger for H2O, which in theabove problem may yield errors as large as several percent.Careful investigation revealed that linear interpolationbetween mole fractions of 0.0 and 0.5 can bring errorsdown to below 0.5% for mole fractions in that range, whileparabolic interpolation using mole fractions of 0.0, 0.5 and1.0 results in errors below 0.1% for the entire mole fractionrange. Since mole fractions of water vapor above 0.5 areunlikely in combustion problems, two mole fractions, 0.0and 0.5, are sufficient for the database.

Since at least 1.5 million wavenumbers are required todescribe the oscillating absorption coefficient for the wholespectrum, and since both the random number and theabsorption coefficient at each wavenumber must be storedin the database, the LBL approach requires a substantialamount of memory. For two species (CO2 and H2O) and23 temperatures, it takes roughly 1.2 GB of memory to

store the whole database in single precision. Keeping inmind that each gas has spectral windows across whichthe absorption coefficient is negligible, the database canbe compressed considerably by storing absorbing bandsonly. While the absorption coefficient of H2O is non-negli-gible across most parts of the spectrum, the absorptionbands of CO2 occupy only a small portion of the spectrum.By adopting this method, the size of the H2O database wasreduced from 768 MB to 657 MB and for CO2 reducedfrom 384 MB to 160 MB. More work can be done toimprove the memory efficiency of the LBL approach, ifdesired. It is also worth noting that, since a constant spec-tral resolution is used for the entire spectrum, fetching datafor a given wavenumber becomes quite efficient since itsindex can be simply calculated.

A gas mixture of 10% CO2 and 20% H2O was employedto validate the constructed species random number andabsorption coefficient database. Emission wavenumbersdetermined from the database using random numbers mustrepresent the distribution of spectral emissive energy of themixture. In Fig. 3a the solid curve is the exact distributionof the normalized spectral emissive energy in spectral inter-vals (dg = 0.01 cm�1), the integrant in Eq. (6), over a smallwavenumber range, while the dashed curve is the corre-sponding Monte Carlo result using 10 million randomnumbers for the entire spectrum. The Monte Carlo resultfollows the exact curve with small statistical errors, whichwill further diminish if the number of random numbers isincreased. In Fig. 3b the mixture absorption coefficient asinterpolated from the proposed database lies on top ofthe curve calculated directly from HITEMP and CDSD.

In contrast, the FSK approach is much more efficient interms of memory. Since change of concentrations maygreatly affect the accuracy of a k-distribution, not less than10 mole fractions for each species are required in building

Wavenumber, cm-1 Wavenumber, cm-1

Abs

orpt

ion

coef

ficie

nt, c

m-1

bar-1

2311.4 2311.6 2311.8 2312 2312.2 2312.4

2

4

6

8 0.00.10.20.51.0

p= 1bar, T=1600K xCO2

1607 1607.5 1608

0.01

0.02

0.03

0.04

0.00.10.20.51.0

xH2O p= 1bar, T= 1600K

Abs

orpt

ion

coef

ficie

nt, c

m-1

bar-1

a b

Fig. 2. Pressure-based absorption coefficient in a small spectral interval (obtained from Wang and Modest [27]): (a) CO2 and (b) H2O.


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the FSK random-number database. For 2 species, 23 tem-peratures and 11 mole fractions, the total arrays of randomnumbers, k-values and stretching factors require onlyabout 4 MB, if as many as 64 points are stored for each dis-tribution. In the present work FSK’s are assembled from ahigh-accuracy compact narrow-band k-distribution data-base developed by Wang and Modest [28], to avoid theinefficient direct assembly from absorption coefficients.Unlike the LBL database, which can be constructed onceand for all, the FSK database depends on the referencestate of the gas mixture and must be constructed for theproblem at hand, and must be reconstructed wheneverthe reference state is changed appreciably.

4. Sample calculations

The PMC/LBL and PMC/FSK implementations arecompared in several sample calculations. First, a probleminvolving a 2D axisymmetric homogeneous CO2–H2O–N2

mixture, surrounded by cold black walls, is considered.The gas temperature is set to a uniform 1620 K and molefractions of CO2 and H2O of 1.4% and 2.8%, respectively,are considered (i.e., the reference state of a methane/airflame discussed later in this section). The radius and lengthare set to R = 25 cm and L = 100 cm. Both the PMC/LBLand PMC/FSK methods are implemented in an energy-partitioning Monte Carlo solver designed for 2D axisym-metric continuum media as posted on the website ofModest’s text [29]. The side wall is divided into four equal-sized rings, which are numbered together with the two endsurfaces as shown in Fig. 4, and radiative fluxes are inves-tigated using both PMC/LBL and PMC/FSK methods.For each run, a certain total number (Nray) of photon raysis traced and each ray carries a number (Nwv) of wavenum-bers (or g’s) randomly determined using the discussed ran-dom-number databases. A 64-bit 2.4 GHz AMD Opteron

880 CPU was used for all the simulations in the presentpaper.

For given numbers of rays and wavenumbers (or g’s) perray, the standard deviations of averaged wall fluxes areevaluated from 50 runs using different random-numbersequences. The purpose of using such a large number ofruns is to accurately evaluate standard deviations and theaveraged CPU time per run. The averaged wall flux usedhere is defined as

�q ¼ 1

A

ZA

q � ndA; ð14Þ

where q is the local heat flux vector at the wall, A is the wallarea and n is its surface normal. Table 1 lists the standarddeviations r (relative to the mean value h�qi) correspondingto each wall flux as well as the averaged simulation time fora single run, using either the LBL or the FSK approach.For each averaged wall flux �q, the relative standard devia-tion is defined as

r ¼ 1

h�qi1

N � 1

XN

n¼1

ð�qðnÞ � h�qiÞ2" #1=2

and

h�qi ¼ 1

N

XN

n¼1

�qðnÞ; ð15Þ

where N = 50 is the total number of runs and �qðnÞ is theaveraged wall flux in the nth run. The mean values of aver-

Wavenumber, cm-1

Ab

sorp

tio

n c

oef

fici

ent,

cm

-1

4140 4142 4144 4146 4148 4150

10-4

10-3

10-2 Direct calculationInterpolation

p= 1bar,T =1200 KxCO2=0.1 ,x H2O=0.2

Fig. 3. Normalized spectral emissive energy and pressure-based absorption coefficient in small spectral intervals at different mole fractions.

r

(1)

(6)(5)

(4)(3)(2)

zo

Fig. 4. Wall numbering.


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aged wall fluxes are h�q1i ¼ h�q4i ¼ 1:41 W=cm2, h�q2i ¼h�q3i ¼ 1:65 W=cm2 and h�q5i ¼ h�q6i ¼ 1:33 W=cm2, using200,000 rays for both LBL and FSK approaches, sincethe FSK approach is exact for homogeneous media. Differ-ent values of Nray are used in simulations to investigate theeffects of number of rays per run, and different values ofNwv are tried with the expectation that more wavenumbersper ray might yield smaller statistical errors without signif-icant increase in CPU time. In nongray analyses the statis-tical error results from spatial uncertainty, due to randomemission locations and directions, and from spectral uncer-tainty due to random wavenumbers (or g’s) approximatingthe spectrum. It is observed that, while standard deviationsare inversely proportional to the square-root of the numberof rays Nray as expected, effects of the number of wavenum-bers Nwv on standard deviations are minor, which impliesthat spatial uncertainty is the major cause for the statisticalerror, and that the spectral variation of radiative intensitiescan be well resolved even with as few as 30,000 photonbundles per run in this problem. As a result, the combina-tion of 120,000 rays per run with 1 wavenumber per ray hasonly one half the standard deviation as compared to thecombination of 30,000 rays per run with 4 wavenumbersper ray, with a similar cost of CPU time. Table 1 showsthat both LBL and FSK approaches achieve the same levelof statistical error. It appears that the PMC/LBL approachdoes its own spectral reordering, given by Eq. (6), in effectsimilar to the FSK reordering. The FSK approach is muchmore efficient in terms of CPU time as expected, since spa-tial calculations are fairly simple and spectral calculationsdominate the cost of CPU time in this homogeneous prob-lem. Even though the LBL spectrum contains 100,000times as many data points as a full-spectrum k-distribution,the LBL approach is only 3–4 times slower than the FSKapproach for given values of Nray and Nwv as shown in Ta-ble 1. This is expected, since a bisectional search algorithmis used in database queries. For an ordered table containingn entries, the average number of comparisons using a bisec-tional search algorithm is log2n. Therefore, for a LBL spec-

trum containing 1.5–3 million wavenumbers, the averagenumber of data comparisons per query is around 21, while,for a 64-point k-distribution, the average number of com-parisons is 6 per query.

To test the PMC/LBL and PMC/FSK methods in inho-mogeneous media, the medium in the previous problem isreplaced by a two-layer medium with step changes of tem-perature and concentrations in the radial direction. A hotcore (r 6 R/2) at 1600 K contains 10% of CO2 and 5% ofH2O, surrounded by a cold layer (R/2 < r 6 R) at 500 Kcontaining 5% of CO2 and 10% of H2O, respectively. Thecorrelation of absorption coefficient is expected to breakdown in such problems with substantial gradients in tem-perature and/or concentrations. The uncorrelatedness ofemission and absorption states results in a larger numberof spectral random numbers required to resolve both emis-sion and absorption, as shown in Table 2. For a givencombination of Nray and Nwv, standard deviations in theinhomogeneous problem are noticeably greater than thosein the homogeneous problem. As a result, unlike in thehomogeneous problem, increasing Nwv also decreases the

Table 1Relative standard deviations of wall fluxes and averaged CPU time per simulation; 50 runs; homogeneous medium problem

Nray Nwv Relative standard deviations (%) tcpu (s)

r1 r2 r3 r4 r5 r6

(a) LBL approach

30,000 1 1.47 1.54 1.39 1.42 1.96 2.15 0.3374 1.51 1.29 1.38 1.32 2.02 1.87 1.325

60,000 1 0.898 0.948 0.852 0.915 1.40 1.46 0.6974 0.954 0.831 0.981 0.957 1.50 1.35 2.577

120,000 1 0.710 0.700 0.589 0.716 0.954 1.13 1.358

(b) FSK approach

30,000 1 1.33 1.25 1.28 1.36 1.69 2.21 0.0934 1.05 1.15 1.29 1.07 1.95 2.36 0.258

60,000 1 1.04 0.915 0.832 1.075 1.31 1.51 0.1884 0.831 0.752 0.897 0.882 1.32 1.39 0.517

120,000 1 0.764 0.642 0.600 0.778 1.07 1.21 0.377

Nray, number of rays per run; Nwv, number of wavenumbers per ray; tcpu, average CPU time per run.

Table 2Relative standard deviations of wall fluxes in the inhomogeneous mediumproblem (50 runs)

Nray Nwv Relative standard deviations (%)

r1 r2 r3 r4 r5 r6

(a) LBL approach

30,000 1 2.35 2.05 2.14 2.28 2.22 2.464 1.21 1.58 1.43 1.36 2.03 2.04

60,000 1 1.77 1.55 1.29 1.78 1.68 1.914 0.900 1.13 1.08 1.17 1.53 1.47

120,000 1 1.31 1.09 0.922 1.04 1.22 1.26

(b) FSK approach

30,000 1 1.98 1.67 1.77 2.12 2.33 2.934 1.46 1.39 1.42 1.57 2.12 2.52

60,000 1 1.54 1.24 1.25 1.41 1.72 1.944 1.01 0.866 0.877 1.19 1.50 1.45

120,000 1 1.12 0.932 1.02 1.08 1.26 1.47


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standard deviation. However, the increase of Nwv is still notas effective as increasing Nray, since the latter reduces bothspatial and spectral uncertainties at the same time. TheLBL approach tends to yield a higher standard deviationthan the FSK approach, since similar random numbersmay be inverted to very different emission wavenumbers,which results in quite different absorption during ray-trac-ing, while in the FSK method similar random numbers areinverted to similar g-values, resulting in similar absorption,due to the correlated-k assumption. However, such effectsare hard to notice here since spectral uncertainty is onlya minor contributor to the standard deviation. The sameray-tracing algorithm and finite-volume mesh used in thehomogeneous problem were employed here and, thus, thecomputational times were the same.

While both LBL and FSK approaches yield comparablestatistical errors, they are not equivalent in terms of accu-racy. Since the LBL approach always uses exact localabsorption coefficients in both emission and absorptioncalculations, the exact result can be achieved if a sufficientnumber of rays are traced. In contrast, the FSK approachuses approximated k-values based on the correlated-kassumption, which breaks down in this inhomogeneousproblem, and results in significant errors compared toLBL results, as shown in Table 3. The FSK approachover-predicts total emission and averaged wall fluxes byaround 10% in this problem.

To test the performance of PMC/LBL and PMC/FSKmethods in more realistic scenarios, time-averaged cell-mean fields are obtained from the composition-PDF solu-tion of a methane/air axisymmetric jet flame after statisticalconvergence has been achieved, and fed into the previousPMC solver. The flame is a scaled up Sandia Flame D

[30] and has a central jet of methane and air mixed witha volume ratio of 1.3 at 293 K flowing into an open spaceat 24.8 m/s and the outer coflow of air is at 291 K and0.45 m/s. Around the fuel jet an annular pilot flow of burntgas enters at 1880 K and 5.7 m/s and the composition ofthe pilot gas is the equilibrium composition for burningof methane with an equivalence ratio of 0.77. The diameterof the fuel jet is D = 14.4 mm and the outer diameter of thepilot flow annulus is 2.62D. The same computationaldomain and finite-volume mesh used in the homogeneousand the step-change problems are used again in the presentproblem. The P1-approximation in conjunction with theFSK model was used to solve radiative transfer for thecomposition-PDF method [30]. Fig. 5 shows the time-aver-aged temperature and concentration fields. A one-stepreaction mechanism is adopted in the PDF model and, asa result, the mole fraction ratio of products, water vaporand carbon dioxide, is constant everywhere and the uncor-relatedness of absorption coefficients is solely due to tem-perature variations. Again, the LBL and FSK approachesyield roughly the same statistical errors and 30,000 photonbundles per run are sufficient to resolve the spectral varia-tion of radiative intensities, as shown in Table 4. Table 5compares the LBL and FSK approaches in terms of totalemission and boundary fluxes. FSK results are not exactagain but errors are much smaller here than in the step-change case, which coincides with the observations ofZhang and Modest [23,24] as mentioned earlier.

It was also observed that the FSK approach is 3–4 timesfaster than the LBL approach, similar to the homogeneouscase. However, the FSK approach requires additional timeto prepare the random-number relations for all possiblestates after the reference state is determined. Since the

Table 3Total emission and wall fluxes in the step-change problem; 50 � 2,000,000 rays and one wavenumber (or g) per ray

Emission (W) Averaged wall fluxes (W/cm2)

�q1 �q2 �q3 �q4 �q5 �q6

LBL 54,747 0.646 0.790 0.790 0.646 0.946 0.946FSK 62,075 0.714 0.875 0.875 0.714 1.04 1.04

Error (%) 13.4 10.5 10.8 10.8 10.5 9.94 9.94

z/D

r/D

0 10 20 30 40 50 60 700

4

8

T(K): 400 600 800 1000 1200 1400 1600 1800 2000

z/D

r/D

0

4

8

xCO2=0.5xH2O: 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 10 20 30 40 50 60 70

Fig. 5. Time-averaged cell-mean fields of temperature and concentrations.


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typical computational times for assembling a full-spectrumk-distribution from the narrow-band k-distribution data-base is 0.1 s, for as few as 20 temperatures and 10 molefractions for each of two species, setup time will require aminimum of 200 s.

Up to this point, PMC/LBL and PMC/FSK methodshave been tested in continuum media using traditionalray-tracing methods. As mentioned earlier, media repre-sented by discrete particle fields are also frequently encoun-tered in combustion simulations using the PDF method.The PDF method was first developed by Pope [12] to treatchemical sources in turbulent reacting flows. Fluctuatingscalars are treated as random variables and a differentialequation predicting the probability-density-function forthese scalars is generated. Because of its high dimensionalitysolutions to the PDF are found by a stochastic Monte Carlo

approach, in which the fluid is modeled by a sufficientlylarge collection of fluid particles (point masses). Any turbu-lent quantity can then be evaluated exactly as long as it is afunction of local scalars only (such as temperature and spe-cies concentrations). Wang and Modest [14] developedthree absorption schemes, the Cone-PPM, Cone-SPM andLine-PPM schemes, by assigning a conical influence regionto rays and/or a spherical influence region to particles, forray-tracing in such media. All three absorption schemesare equivalent in terms of simulation accuracy. They alsoproposed an adaptive emission scheme to reduce statisticalerrors of PMC simulations, by splitting high particle ener-gies and combining low particle energies to keep ray ener-gies relatively constant [15]. The present PMC/LBL andPMC/FSK methods have also been incorporated into theabove absorption and emission schemes to deal with mediarepresented by particle fields. In the present paper the Cone-PPM scheme is adopted for sample calculations and the rec-ommended value of a 1� ray opening angle is used.

A snapshot of a particle field containing roughly 80,000stochastic particles is extracted from the particle MonteCarlo solution of the composition-PDF simulation of thescaled Sandia D flame [30], which was already used in theprevious example. The turbulence scale in a turbulent jetflame depends on the properties of the fuel jet and the Rey-nolds number based on the fuel jet is essential in character-izing turbulent flows. In the present flame the Reynoldsnumber is 22,400. The scale of the radiation field is gener-ally characterized by the optical thickness, which is calcu-lated based on the Planck-mean absorption coefficient ofthe combustion products and the flame length. The opticalthickness of the present flame is 0.474. Fig. 6 shows the cell-mean values of temperature and CO2 mole fraction. Again,because of the one-step reaction mechanism used in the

Table 4Relative standard deviations obtained in time-averaged cell-mean fields(50 runs)

Nray Nwv Relative standard deviations (%)

r1 r2 r3 r4 r5 r6

(a) LBL approach

30,000 1 1.59 1.18 1.09 0.97 2.08 1.424 1.12 1.01 0.91 1.27 1.98 1.59

60,000 1 1.11 0.834 0.773 0.837 1.27 1.064 1.02 0.659 0.650 1.04 1.69 1.07

120,000 1 0.691 0.532 0.450 0.557 1.02 0.73

(b) FSK approach

30,000 1 1.34 1.04 1.17 1.06 1.96 1.664 1.50 1.03 0.89 1.32 2.10 1.49

60,000 1 0.982 0.697 0.685 0.732 1.64 0.9974 1.01 0.765 0.654 0.941 1.62 0.950

120,000 1 0.621 0.530 0.457 0.586 1.14 0.785

Table 5Total emission and boundary fluxes in time-averaged cell-mean fields; 50 � 2,000,000 rays and one wavenumber (or g) per ray

Emission (W) Averaged wall fluxes (W/cm2)

�q1 �q2 �q3 �q4 �q5 �q6

LBL 3513 0.0580 0.104 0.116 0.0775 0.0646 0.0753FSK 3589 0.0582 0.105 0.117 0.0771 0.0641 0.0733

Error (%) 2.16 0.34 0.96 0.86 �0.52 �0.77 �2.66

z/D

r/D

0

4

8

T(K): 400 600 800 1000 1200 1400 1600 1800 2000

z/D

r/D

0

4

8

xCO2=0.5xH2O: 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70

Fig. 6. Cell-mean contours of temperature and concentrations calculated from a snapshot of particle fields.


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simulation, the ratio of mole fractions of CO2 and H2O isconstant everywhere. The roughness of cell-mean fieldsimplies large local gradients of temperature and mole frac-tions. Fig. 6 is only a single snapshot of cell-mean values,which fluctuate with time indicating a turbulent medium.By time-averaging over all snapshots, the time-averagedcell-mean fields can be obtained as given in Fig. 5. Interac-tions between turbulence and radiation tend to enhance theheat loss. Effects of turbulence–radiation interactions(TRI) may be investigated with such ‘‘frozen” particle field.In treatments of TRI using deterministic methods, only thecoupling between radiative emission and local fluctuatingabsorption coefficients, ‘‘emission TRI”, can be considered,and the coupling between incoming radiative intensitiesand local absorption coefficients, ‘‘absorption TRI”, isignored by applying the so-called optically-thin-fluctuationassumption (OTFA) [31]. However, in joint-PDF methods,the instantaneous scalar fields are known and representedby particle fields evolving with time, and the TRI can befully calculated if a photon Monte Carlo simulation is usedto determine radiation. If properties of the individual sto-chastic particle (temperature and absorption coefficient)are employed in both emission and absorption calculations,both emission TRI and absorption TRI are taken intoaccount, labeled here ‘‘full-TRI.” If properties of the indi-vidual stochastic particle are used in emission calculations,but cell-mean values are used in absorption calculations,then only the emission TRI is taken into account (equiva-lent to the OTFA assumption), called here ‘‘partial-TRI.”If cell-mean values are used in both emission and absorp-tion calculations, turbulence–radiation interactions areneglected entirely.

Relative standard deviations of boundary fluxes and theaveraged CPU time are compared in Table 6 for the LBLand FSK approaches using various amounts of rays perrun and wavenumbers per ray. It is observed that 30,000rays per run are sufficient to resolve the spectral uncer-tainty and the spacial uncertainty dominates the statisticalerror once again for both LBL and FSK approaches in this

turbulent particle field. However, if the cell-based heatsource terms are desired, as required in the energy equationin combustion modeling, a much higher number of rayswill be needed to achieve the same level of statistical errorin terms of heat source, considering that the scale of finite-volume cells is generally much smaller than that of the sixsurfaces under investigation here. Unlike previous cases,the difference in CPU-time efficiency between the LBLand FSK approaches is not substantial, because the ray–particle interaction is the most time-intensive part in theMonte Carlo simulations in particle fields, making the dif-ference between spectral models less prominent. Table 7shows that the FSK approach over-estimates the totalemission from the particle field by about 2% and under-estimates the total heat loss by 6% due to the inaccuracyof the correlated-k assumption, which leads to an over-esti-mation of absorption (emission minus heat loss) by around16%. The gray analysis, on the other hand, in which localPlanck-mean absorption coefficients are used in both emis-sion and absorption calculations, over-estimates the totalheat loss by over 50%.

Effects of turbulence–radiation interactions were alsoinvestigated for the above particle field. Table 8 lists thetotal emission and total heat loss for the three differentTRI treatments. The PMC/LBL method was employed inall cases. It is observed that total emission is under-esti-mated by 14% if TRI is neglected, showing that emissionTRI must be taken into account in the present turbulentjet flame. However, there is only a negligible difference

Table 6Relative standard deviations of wall fluxes and averaged CPU time per simulation; 50 runs; particle field

Nray Nwv Relative standard deviations (%) tcpu (s)

r1 r2 r3 r4 r5 r6

(a) LBL approach

30,000 1 1.56 1.09 0.923 1.25 2.28 1.61 13.44 1.22 1.20 0.979 1.20 2.29 1.62 19.2

60,000 1 1.11 0.826 0.796 0.918 1.69 1.31 27.74 1.11 0.871 0.757 0.782 1.37 0.870 36.6

120,000 1 0.776 0.627 0.558 0.635 1.16 0.993 48.3

(b) FSK approach

30,000 1 1.84 1.44 1.13 1.40 2.30 1.97 10.74 1.97 1.24 1.09 1.56 2.38 2.65 13.6

60,000 1 1.28 0.720 0.743 0.987 1.96 1.52 21.04 1.17 0.987 0.806 1.06 1.95 1.31 26.0

120,000 1 0.751 0.652 0.577 0.678 1.23 1.06 42.9

Table 7Total emission and heat loss from particle field for different spectralmodels (in W; 50 � 200,000 rays)

Emission Heat loss

Mean Error (%)

LBL 4920 3029 –FSK 5034 2847 �6.0Gray 4920 4589 51.5


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pybetween the full-TRI and the partial-TRI treatments in theheat loss estimation, which implies that absorption TRI isnegligible and the OTFA assumption is valid in the presentflame model. This was to be expected since, though absorp-tion TRI may be appreciable in some small parts of thespectrum where absorption coefficients are large, theoverall effect is still expected to be minimal after integra-tion over the entire spectrum, as pointed out by Hartick[32].

From the above sample calculations one observes thatboth PMC/LBL and PMC/FSK approaches result inroughly the same level of statistical errors in both homoge-neous and inhomogeneous media for the same number ofrays. By using the random-number–wavenumber relation,the LBL method does its own spectral reordering, in effectsimilar to the FSK reordering. The spectral behavior canbe resolved with a limited number of rays while spacialuncertainty is the major contributor to statistical errors.The FSK approach is superior in CPU-time efficiency overthe LBL approach, but this advantage is diminished inmedia represented by particle fields due to the time-inten-sive ray-tracing process. Because of the time-consumingdatabase preparation the FSK approach also becomes lessCPU efficient in quickly-evolving media.

5. Summary

Spectral LBL and FSK approaches have been incorpo-rated into PMC models to calculate radiative heat transferin nongray combustion gases. Random-number relationsare formulated for gaseous mixtures, which can be calcu-lated efficiently based on individual species random-num-ber relations, and such databases were constructed forwater vapor and carbon dioxide, the two most importantcombustion gases, using pressure-based absorption coeffi-cients. While much more memory efficient than the LBLapproach, the FSK method is not promising in terms ofCPU-time efficiency, since the calculational effort of theFSK algorithm increases exponentially with the numberof species, while that of the LBL algorithm increases onlylinearly. Considering the time-consuming database prepa-ration process, the FSK method is not suitable forquickly-evolving media. In contrast, the LBL method doesnot require such a preparation process since the LBL data-base can be constructed once and for all.

By using random-number relations, the wavenumberscontributing the most to radiation have higher probabili-

ties to be chosen during PMC simulations, which is equiv-alent to spectral reordering in the FSK method. Thus, itwas observed that both the LBL and FSK approaches pro-duce the same level of statistical errors for a given numberof photon bundles in various sample cases, including botha homogeneous problem and inhomogeneous problems,and both continuum media and a medium represented bydiscrete particle fields. A relatively small number of rayscan resolve the spectral uncertainty in PMC simulationsfor both LBL and FSK methods. However, the FSKmethod has inherent inaccuracy in inhomogeneous mediasince absorption coefficients are not truly correlated. Tur-bulence–radiation interactions in a methane/air jet flamewere also investigated using the proposed PMC/LBLmethod. It is noted that the PMC method for stochasticmedia is the only method that allows the full evaluationof TRI, including the so-called absorption TRI.

Acknowledgment

This research has been sponsored by National ScienceFoundation under Grant Number CTS-0121573.

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Table 8Total emission and heat loss from particle field for different TRItreatments(in W; 50 � 200,000 rays)

Emission Heat loss

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Full-TRI 4920 3029 –Partial-TRI 4920 3019 �0.33No-TRI 4270 2471 �18.4


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